Bonanno-Reuter Quantum Black Hole

• The Bonanno-Reuter black hole is a quantum-corrected solution that employs renormalization group improvements to replace the classical singularity with a regular de Sitter core.
• It yields observable modifications in perturbation spectra and quasinormal modes, with significant shifts in oscillation frequencies and damping rates due to quantum corrections.
• The model smoothly interpolates between quantum and classical regimes, recovering Schwarzschild behavior in the IR while introducing parameters that control dynamic black hole evolution.

The Bonanno–Reuter black hole is a class of regular, quantum-corrected black hole solutions constructed through renormalization group (RG) improvement of the Schwarzschild solution within the asymptotically safe gravity program. These solutions provide a concrete framework to incorporate RG-induced running of gravitational couplings into classical General Relativity, leading to the resolution of the central singularity by a regular de Sitter core and introducing observable quantum corrections in dynamics, perturbation spectra, and Hawking evaporation.

1. Construction in the Asymptotic Safety Framework

The Bonanno–Reuter approach is grounded in Weinberg’s asymptotic safety scenario, in which the gravitational couplings become scale-dependent via their RG flow (Saueressig et al., 2015). The effective action is approximated as

$$\Gamma_k^{(\mathrm{grav})} = \frac{1}{16\pi G_k} \int d^4x \sqrt{g} (2\Lambda_k - R)$$

where Newton’s constant $G_k$ and the cosmological constant %%%%1%%%% depend on the RG scale $k$. Near the non-Gaussian fixed point (NGFP), the RG flow is governed by: $$G_k = g_* k^{-2} \,,\qquad \Lambda_k = \lambda_* k^2$$ with $g_*$, $\lambda_*$ positive constants. In the IR ($k\to 0$), $G_k \rightarrow G_0$, the standard Newton constant. The “RG improvement” proceeds by promoting $G_0$ in the Schwarzschild lapse

$f(r) = 1 - \frac{2G_0 M}{r}$

to $G(k(r))$, with $k$ identified with a physically meaningful inverse length. The identification $k(r) = \xi/d_r(r)$, where $d_r(r)$ is a modified proper distance to $r$, leads to a quantum-corrected metric of the form: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1} dr^2 + r^2 d\Omega^2, \qquad f(r) = 1 - \frac{2 M G(k(r))}{r}$$ Specific RG trajectories and identification schemes yield models in which the spacetime interpolates smoothly between classical and quantum regimes.

2. Metric Structure and Singularity Resolution

A key feature of the Bonanno–Reuter solution is singularity avoidance via a de Sitter core. For small $r$ (the deep UV), with $G_k \sim g_* k^{-2}$ and $k \rightarrow \xi/d_r(r)$, the lapse behaves as: $$f(r) \approx 1 - \frac{1}{3}\Lambda_\mathrm{eff} r^2\,,\quad \Lambda_\mathrm{eff} = \frac{4g_*}{3G_0 \xi^2}$$ This regularizes classical curvature invariants such as the Kretschmann scalar, which remain finite everywhere. The metric can also be written as $f(r) = 1 - 2M(r)/r$, with $M(r)$ varying smoothly from $0$ at $r=0$ to $M$ at large $r$, matching the Hayward geometry for regular black holes. Thus, the classical singularity at $r=0$ is replaced by a regular region with effective cosmological constant, closely paralleling Planck star phenomenology.

3. Quasinormal Modes and Dynamical Signatures

The RG-improved geometry introduces distinctive features in perturbation spectra (Konoplya et al., 2022, Bolokhov et al., 9 Jul 2025, Malik, 8 Oct 2025). The wave equation for perturbations is

$$\frac{d^2\Psi}{dr_*^2} + [\omega^2 - V(r)]\Psi = 0, \quad dr_*/dr = 1/f(r)$$

For gravitational and scalar perturbations, the effective potential $V(r)$ depends on the modified $f(r)$ (and its derivatives), affecting both oscillation frequencies and damping rates:

• Fundamental quasinormal modes ($n=0$): Deviate weakly from Schwarzschild values for large $M$, recovering classical predictions in this limit. For smaller masses or larger quantum corrections, there are significant shifts in both $\Re(\omega)$ and $|\Im(\omega)|$.
• Overtones ($n\geq 1$): Exhibit strong sensitivity to the near-horizon quantum corrections: fractional deviations reach hundreds of percent. The appearance of almost purely imaginary (non-oscillatory) modes is a hallmark of the quantum–corrected spacetime (Konoplya et al., 2022, Bolokhov et al., 9 Jul 2025).
• Massive scalar fields: Increasing field mass $\mu$ further suppresses the damping, resulting in the emergence of quasi-resonances (arbitrarily long–lived oscillations) (Malik, 8 Oct 2025). Late-time decay for massive fields follows an asymptotic form $\Psi(t) \sim t^{-7/8}\sin(\mu t + \phi)$, distinctly different from massless power-law tails.

4. Hawking Radiation and Grey-Body Factors

Hawking radiation in the Bonanno–Reuter spacetime is characterized by a modified temperature,

$T_H = \frac{f'(r_H)}{4\pi}$

where $r_H$ is the horizon radius. Due to the running $G(r)$, $T_H$ is significantly reduced compared to Schwarzschild with the same mass (Konoplya, 2023). Additionally, the effective potential is raised by quantum corrections, resulting in smaller grey-body factors $\Gamma_\ell(\omega)$, i.e., the transmission probabilities for partial waves are suppressed: $$\Gamma_\ell(\omega) = \left[1+\exp(2\pi K)\right]^{-1},\qquad K \approx \frac{i(\omega^2-V_0)}{\sqrt{-2V_0''}}$$ For massless test fields (e.g., electromagnetic, Dirac), the combined suppression from both lower temperature and reduced transmission leads to a decrease in the Hawking emission rate by several or even many orders of magnitude, irrespective of the specific $k(r)$ identification. For massive scalar fields, the grey-body factors decrease with increasing mass, suppressing low-frequency emission and shifting the radiation spectrum toward higher frequencies (Malik, 8 Oct 2025).

5. Model Parameters and the Classical Limit

The RG-improvement incorporates phenomenological parameters:

• $\tilde{\omega}$: Fixed by matching one-loop quantum corrections.
• $\gamma$ (interpolation parameter): Controls the interpolation between the UV (quantum-dominated) and IR (classical) regime (Bolokhov et al., 9 Jul 2025). Small $\gamma$ enhances quantum corrections near the core, while large $\gamma$ reduces deviations, yielding rapid recovery of the Schwarzschild limit as either $M$ or $\gamma$ increases.
• $\xi$ (scale parameter): Relates the RG scale $k$ to the spacetime point.

A table illustrates the dependency:

Parameter	Physical Role	Effect (for fixed $M$)
$\tilde{\omega}$	1-loop RG correction coeff.	Sets overall quantum strength
$\gamma$	Classical/quantum interpolation	Large $\gamma$: classical behavior; small $\gamma$: strong quantum effects
$\xi$	RG scale-setting	Affects scale of corrections

In all cases, the classical Schwarzschild limit is robustly restored for large $M$ or large $\gamma$.

6. Matter Coupling and Gravitational Collapse

Recent extensions incorporate dynamical matter sources. Within the effective action,

$$S = \frac{1}{16\pi G_N} \int d^4x \sqrt{-g} [R + 2\chi(\epsilon) \mathcal{L}]$$

where the matter Lagrangian couples through a multiplicative function $\chi(\epsilon)$, itself fixed by RG flow and the Reuter fixed point (Bonanno et al., 2023). The associated field equations lead to energy-density–dependent $G(\epsilon)$ that vanishes at high densities, enforcing singularity avoidance even during dust collapse. Matching the interior (collapsing matter) to the exterior static Bonanno–Reuter geometry via Israel junction conditions yields a globally regular black hole spacetime.

7. Phenomenological Implications and Observational Prospects

Bonanno–Reuter black holes provide a regular, quantum-corrected arena for black hole physics and quantum gravity phenomenology. Major implications and features include:

• Regularization of curvature invariants and elimination of classical singularities.
• Quantum-improved QNM and wave propagation spectra, with overtones and massive-field modes offering sensitive probes of near-horizon quantum corrections.
• Distinctive Hawking evaporation profiles: strongly suppressed overall flux and a high-frequency spectral shift, both for massless and massive fields.
• Smooth recovery of general relativistic behavior in the appropriate limits.
• Model independence of qualitative features with respect to the $k(r)$ identification scheme.

These signatures position the Bonanno–Reuter model as a central benchmark for testing quantum gravity modifications against gravitational wave and electromagnetic observations, with overtones and radiation spectra serving as particularly sensitive diagnostics of quantum corrections (Bolokhov et al., 9 Jul 2025, Malik, 8 Oct 2025, Konoplya et al., 2022, Konoplya, 2023, Bonanno et al., 2023).